To simplify the expression $5(3 - 2i) + 2i(4 + 6i)$, we’ll expand each term and then combine like terms.

Step 1: Expand $5(3 - 2i)$

$5(3 - 2i) = 5 \cdot 3 - 5 \cdot 2i = 15 - 10i$

Step 2: Expand $2i(4 + 6i)$

$2i(4 + 6i) = 2i \cdot 4 + 2i \cdot 6i = 8i + 12i^2$ Since $i^2 = -1$, replace $12i^2$ with $12 \cdot (-1) = -12$: $2i(4 + 6i) = 8i - 12$

Step 3: Combine Terms

Now, add the results from Step 1 and Step 2: $15 - 10i + 8i - 12$

Combine like terms (real parts and imaginary parts):

• Real part: $15 - 12 = 3$
• Imaginary part: $-10i + 8i = -2i$

Final Answer

$3 - 2i$

Would you like a deeper explanation or have any further questions?

Here are five related questions to consider:

1. How would the answer change if $i$ were replaced with a different imaginary unit?
2. What is the geometric interpretation of the complex number $3 - 2i$ on the complex plane?
3. How can we factor expressions involving complex numbers?
4. How do we multiply complex numbers using polar form?
5. What other mathematical properties are unique to complex numbers?

Tip: Always remember that $i^2 = -1$; this substitution is essential in simplifying expressions with complex numbers.